Bleach and bleach fumes are highly toxic to humans, never mind birds! I don't care how much you dilute it, or let it dry, or rinse it. It's still a HARSH chemical!
With so many other safe products out there, why would anyone even take the risk of killing their birds. We all know how sensitive their lungs are.
This is MY opinion, just like Teflon, I WOULDN'T take the risk, especially when I can clean just as well with other natural products less harmful to both humans and birds.